53.1.17 problem 17

Internal problem ID [8451]
Book : Elementary differential equations. By Earl D. Rainville, Phillip E. Bedient. Macmilliam Publishing Co. NY. 6th edition. 1981.
Section : CHAPTER 16. Nonlinear equations. Section 94. Factoring the left member. EXERCISES Page 309
Problem number : 17
Date solved : Monday, January 27, 2025 at 04:01:49 PM
CAS classification : [[_homogeneous, `class A`], _rational, _Bernoulli]

\begin{align*} x y \left (x^{2}+y^{2}\right ) \left ({y^{\prime }}^{2}-1\right )&=y^{\prime } \left (x^{4}+x^{2} y^{2}+y^{4}\right ) \end{align*}

✓ Solution by Maple

Time used: 0.553 (sec). Leaf size: 248

dsolve(x*y(x)*(x^2+y(x)^2)*(diff(y(x),x)^2-1)=diff(y(x),x)*(x^4+x^2*y(x)^2+y(x)^4),y(x), singsol=all)
 

\begin{align*} y &= \frac {\sqrt {\left (c_{1} x^{2}-\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1} x^{2}}}{x \left (c_{1} x^{2}-\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1}} \\ y &= \frac {\sqrt {\left (c_{1} x^{2}+\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1} x^{2}}}{x \left (c_{1} x^{2}+\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1}} \\ y &= \frac {\sqrt {\left (c_{1} x^{2}-\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1} x^{2}}}{x \left (-c_{1} x^{2}+\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1}} \\ y &= -\frac {\sqrt {\left (c_{1} x^{2}+\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1} x^{2}}}{x \left (c_{1} x^{2}+\sqrt {c_{1}^{2} x^{4}+1}\right ) c_{1}} \\ y &= \sqrt {2 \ln \left (x \right )+c_{1}}\, x \\ y &= -\sqrt {2 \ln \left (x \right )+c_{1}}\, x \\ \end{align*}

✓ Solution by Mathematica

Time used: 9.430 (sec). Leaf size: 248

DSolve[x*y[x]*(x^2+y[x]^2)*((D[y[x],x])^2-1)==D[y[x],x]*(x^4+x^2*y[x]^2+y[x]^4),y[x],x,IncludeSingularSolutions -> True]
 

\begin{align*} y(x)\to -\sqrt {-x^2-\sqrt {x^4+e^{4 c_1}}} \\ y(x)\to \sqrt {-x^2-\sqrt {x^4+e^{4 c_1}}} \\ y(x)\to -\sqrt {-x^2+\sqrt {x^4+e^{4 c_1}}} \\ y(x)\to \sqrt {-x^2+\sqrt {x^4+e^{4 c_1}}} \\ y(x)\to -x \sqrt {2 \log (x)+c_1} \\ y(x)\to x \sqrt {2 \log (x)+c_1} \\ y(x)\to -\sqrt {-\sqrt {x^4}-x^2} \\ y(x)\to \sqrt {-\sqrt {x^4}-x^2} \\ y(x)\to -\sqrt {\sqrt {x^4}-x^2} \\ y(x)\to \sqrt {\sqrt {x^4}-x^2} \\ \end{align*}